2.1 Hierarchical Vector Quantization

2.1.3 Quantization Error

Let us try to understand quantization error with an example.

Figure 1: The Voronoi tessellation for level 1 shown for the 5 cells with the points overlayed

An example of a 2 dimensional VQ is shown above.

In the above image, we can see 5 cells with each cell containing a certain number of points. The centroid for each cell is shown in blue. These centroids are also known as codewords since they represent all the points in that cell. The set of all codewords is called a codebook.

Now we want to calculate quantization error for each cell. For the sake of simplicity, let’s consider only one cell having centroid A and m data points \(F_i\) for calculating quantization error.

For each point, we calculate the distance between the point and the centroid.

\[ d = ||A - F_i||_{p} \]

In the above equation, p = 1 means L1_Norm distance whereas p = 2 means L2_Norm distance. In the package, the L1_Norm distance is chosen by default. The user can pass either L1_Norm, L2_Norm or a custom function to calculate the distance between two points in n dimensions.

\[QE = \max_i(||A-F_i||_{p})\]

Now, we take the maximum calculated distance of all m points. This gives us the furthest distance of a point in the cell from the centroid, which we refer to as Quantization Error. If the Quantization Error is higher than the given threshold, the centroid/ codevector is not a good representation for the points in the cell. Now we can perform further Vector Quantization on these points and repeat the above steps.

Please note that the user can select mean, max or any custom function to calculate the Quantization Error. The custom function takes a vector of m value (where each value is a distance between point in n dimensions and centroids) and returns a single value which is the Quantization Error for the cell.

If we select mean as the error metric, the above Quantization Error equation will look like this :

\[QE = \frac{1}{m}\sum_{i=1}^m||A-F_i||_{p}\]

6.1 Execution of Workflow Steps at Level 1

In this section all the outlined workflow steps provided in the abstract section (Compression, Projection and Tessellation) are executed at level 1.

Step 1: Data Compression

The core function for compression in the workflow is HVQ, which is called within the HVT function. we have a parameter called quantization error. This parameter acts as a threshold and determines the number of levels in the hierarchy. It means that, if there are ‘n’ number of levels in the hierarchy, then all the clusters formed till this level will have quantization error equal or greater than the threshold quantization error. The user can define the number of clusters in the first level of hierarchy and then each cluster in first level is sub-divided into the same number of clusters as there are in the first level. This process continues and each group is divided into smaller clusters as long as thethreshold quantization error is met. The output of this technique will be hierarchically arranged vector quantized data.

However, let’s try to comprehend the HVT function first before moving on.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  projection.scale,
  normalize = T,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  diagnose = TRUE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of HVT function have been explained below :

• dataset - A dataframe with numeric columns

• min_compression_perc - An integer indicating the minimum percent compression rate to be achieved for the dataset

• n_cells - An integer indicating the number of cells per hierarchy (layers)

• depth - An integer indicating the number of layers. (1 = No hierarchy, 2 = 2 layers, etc …)

• quant.error - A number indicating the quantization error threshold. A cell will only breakdown into further cells if the quantization error of the cell is above the defined quantization error threshold

• distance_metric - The distance metric can be L1_Norm or L2_Norm. L1_Norm is selected by default. The distance metric is used to calculate the distance between an n dimensional point and centroid. The user can also pass a custom function to calculate this distance

• error_metric - The error metric can be mean or max. max is selected by default. max will return the max of m values and mean will take mean of m values where each value is a distance between a point and centroid of the cell. Moreover, the user can also pass a custom function to calculate the error metric

• quant_method - The quantization method can be kmeans or kmedoids. kmeans is selected by default

• normalize - A logical value indicating whether the columns in your dataset need to be normalized. Default value is TRUE. The algorithm supports Z-score normalization

• diagnose - A logical value indicating whether user wants to perform diagnostics on the model. Default value is TRUE.

• hvt_validation - A logical value indicating whether user wants to holdout a validation set and find mean absolute deviation of the validation points from the centroid. Default value is FALSE.

• train_validation_split_ratio - A numeric value indicating train validation split ratio. This argument is only used when hvt_validation has been set to TRUE. Default value for the argument is 0.8

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce the level of detail captured in the data representation

We will pass the below mentioned model parameters along with torus dataset to HVT function.

Model Parameters

• Number of Cells at each Level = 100
• Maximum Depth = 1
• Quantization Error Threshold = 0.1
• Error Metric = Max
• Distance Metric = Manhattan

set.seed(240)
hvt.torus <- muHVT::HVT(
  torus_df,
  n_cells = 100,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary .

compressionSummaryTable(hvt.torus[[3]]$compression_summary)

As it can be seen from the table above, none of the 100 cells have hit the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

• Number of Cells at each Level = 300
• Maximum Depth = 1
• Quantization Error Threshold = 0.1
• Error Metric = Max
• Distance Metric = Manhattan

set.seed(240)
hvt.torus2 <- muHVT::HVT(
  torus_df,
  n_cells = 300,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary again.

compressionSummaryTable(hvt.torus2[[3]]$compression_summary)

It can be observed from the table above that only 43 cells out of 300 i.e. 14% of the cells hit the Quantization Error threshold. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if 80% compression is reached

Now let’s try again with the below mentioned set of model parameters.

Model Parameters

• Number of Cells at each Level = 900
• Maximum Depth = 1
• Quantization Error Threshold = 0.1
• Error Metric = Max
• Distance Metric = Manhattan

set.seed(240)
hvt.torus3 <- muHVT::HVT(
  torus_df,
  n_cells = 900,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s check the compression summary for torus.

compressionSummaryTable(hvt.torus3[[3]]$compression_summary)

By increasing the number of cells to 900, we were successfully able to compress 93% of the data, so we will not further subdivide the cells

Step 2: Data Projection

The function sammonsProjection() utilizes the sammons function from the MASS package being called in HVT. Sammon’s projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality while attempting to preserve the structure of inter-point distances in the projection.The centroids are plotted in 2D after performing Sammon’s projection at every level of the tessellation.

lets view the projected 2D centroids after performing sammon’s projection. For the shake of brevity we are displaying first six rows.

hvt_torus_coordinates <-hvt.torus3[[2]][[1]][["1"]]
centroids <<- list()
  coordinates_value <- lapply(1:length(hvt_torus_coordinates), function(x){
    centroids <-hvt_torus_coordinates[[x]]
    coordinates <- centroids$pt
  })
centroid_coordinates<<- do.call(rbind.data.frame, coordinates_value)  
colnames(centroid_coordinates) <- c("x","y")
centroid_coordinates <- centroid_coordinates %>% data.frame() %>% round(4)
Table(head(centroid_coordinates), scroll = T, limit = 20)

Step 3: Tessellation

The deldir package computes the Delaunay triangulation (and hence the Dirichlet or Voronoi tessellation) of a planar point set according to the second (iterative) algorithm of Lee and Schacter. For subsequent levels, transformation is performed on the 2D coordinates to get all the points within its parent tile. Tessellations are plotted using these transformed points as centroids.

plotHVT is the main function to plot hierarchical voronoi tessellations.

Now let’s try to understand plotHVT function. The parameters have been explained in detail below

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

• hvt.results - A list containing the output of the HVT function which has the details of the tessellations to be plotted

• line.width - A vector indicating the line widths of the tessellation boundaries for each layer

• color.vec - A vector indicating the colors of the tessellations boundaries at each layer

• pch1 - Symbol type of the centroids of the tessellations (parent levels). Refer points (default = 21)

• centroid.size - Size of centroids of first level tessellations (default = 3)

• title - Set a title for the plot (default = NULL)

For better visualisation, let’s plot the Voronoi tessellation.

muHVT::plotHVT(
  hvt.torus3,
  line.width = c(0.4),
  color.vec = c("#141B41"),
  centroid.size = 0.6,
  maxDepth = 1
)

From the presented plot, the inherent structure of the donut can be easily observed in the two-dimensional space

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data. .

Let’s have look at the hvtHmap function which we will use to overlay a variable as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

• hvt.results - A list of results obtained from the HVT function

• dataset - A dataframe containing the variables to overlay as a heatmap. The user can pass an external dataset or the dataset that was used to perform hierarchical vector quantization. The dataset should have the same number of points as the dataset used to perform hierarchical Vector Quantization in the HVT function

• child.level - A number indicating the level for which the heat map is to be plotted

• hmap.cols - The column number of column name from the dataset indicating the variables for which the heat map is to be plotted. To plot the quantization error as heatmap, pass 'quant_error'. Similarly to plot the no of points in each cell as heatmap, pass 'no_of_points' as a parameter

• color.vec - A color vector such that length(color.vec) = child.level (default = NULL)

• line.width - A line width vector such that length(line.width) = child.level (default = NULL)

• palette.color - A number indicating the heat map color palette. 1 - rainbow, 2 - heat.colors, 3 - terrain.colors, 4 - topo.colors, 5 - cm.colors, 6 - BlCyGrYlRd (Blue,Cyan,Green,Yellow,Red) color (default = 6)

• show.points - A boolean indicating whether the centroids should be plotted on the tessellations (default = FALSE)

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

metric_list <- colnames(hvt.torus3[[3]]$summary)
metric_list <- metric_list[7:9]
hmap <- list()
hmap <- lapply(1:length(metric_list), function(x){
muHVT::hvtHmap(
  hvt.torus3,
  torus_df,
  child.level = 1,
  hmap.cols = metric_list[[x]],
  line.width = c(0.4),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 1,
  show.points = T,
  quant.error.hmap = 0.1,
  n_cells.hmap = 900
)
})

The heatmaps displayed below provides a visual representation of the spatial characteristics of the torus, allowing us to observe patterns and trends in the distribution of each of the features (X,Y and Z). The sheer green shades highlight regions with higher coordinate values in each of the heatmaps, while the indigo shades indicate areas with the lowest coordinate values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the torus structure

grid.arrange(hmap[[1]], nrow = 1, ncol=1)

grid.arrange(hmap[[2]] ,nrow = 1, ncol=1)

grid.arrange(hmap[[3]], nrow = 1, ncol=1)

6.2 Refining Torus Projection at Level 2

In this section we will explore a little more and try to project the torus(3D) object to 2D Space with different set of model parameters at level 2 inorder to achieve more refined and detailed tessellation

Through this exploration, we can gain a deeper understanding of the transformation of the torus object into a 2D projection. By adjusting the model parameters specifically for the second level of tessellation, we expect to achieve a higher level of refinement and granularity in the resulting projection, enabling us to analyze the torus in greater detail and extract meaningful insights from the transformed data.

Step 1: Data Compression

First we will perform Hierarchical Vector Quantization using the torus data along with below mentioned model parameters to achieve a compression summary of atleast 80%.

for detailed information on data Compression please refer to section 2 of this vignette.

Model Parameters

• Number of Cells at each Level = 20
• Maximum Depth = 2
• Quantization Error Threshold = 0.1
• Error Metric = Max
• Distance Metric = Manhattan

set.seed(240)
hvt.torus4 <- muHVT::HVT(
  torus_df,
  n_cells = 20,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary for torus and see whether the model has achieved 80% compression or not.

compressionSummaryTable(hvt.torus4[[3]]$compression_summary)

It can be observed from the table above that none of the cells has hit the Quantization Error threshold. Therefore we can further subdivide the cells by increasing the n_cells inorder to achieve a compression of 80%

Since we are not able to achieve our compression objective. Let’s try again with the below mentioned set of model parameters.

Model Parameters

• Number of Cells at each Level = 26
• Maximum Depth = 2
• Quantization Error Threshold = 0.1
• Error Metric = Max
• Distance Metric = Manhattan

set.seed(240)
hvt.torus5 <- muHVT::HVT(
  torus_df,
  n_cells = 26,
  depth = 2,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Now,let’s checkout the compression summary for torus by.

compressionSummaryTable(hvt.torus5[[3]]$compression_summary)

It can be observed from the table above that only 535 cells out of 666 i.e. 80% of the cells hit the Quantization Error threshold. Since we have attained 80% compression we will not further subdivide the cells

Step 2: Data Projection

lets view the projected 2D centroids after performing sammon’s projection on the compressed data recieved after performing vector quantization.

For detailed information on Data Projection please refer to section 3 of this vignette.

For the shake of brevity we are displaying first six rows.

hvt_torus_coordinates <-hvt.torus5[[2]][[1]][["1"]]
centroids <<- list()
  coordinates_value <- lapply(1:length(hvt_torus_coordinates), function(x){
    centroids <-hvt_torus_coordinates[[x]]
    coordinates <- centroids$pt
  })
centroid_coordinates<<- do.call(rbind.data.frame, coordinates_value)  
colnames(centroid_coordinates) <- c("x","y")
centroid_coordinates <- centroid_coordinates %>% data.frame() %>% round(4)
Table(head(centroid_coordinates), scroll = T, limit = 20
             )

Step 3: Tessellation

Now, we have obtained the centroid coordinates resulting from the application of Sammon’s projection.

For better visualisation, let’s plot the Voronoi tessellation using the plotHVT function.

muHVT::plotHVT(
  hvt.torus5,
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  centroid.size = 0.8,
  maxDepth = 2
)

From the presented plot, the inherent structure of the donut can no longer be observed in the two-dimensional space

NOTE

When we pass the 3D torus data to the HVT function with depth 1, it performs a variance decomposition at a single level. This means that the algorithm calculates the overall variance of the data and provides a 2D visualization that represents the variance structure in a simplified manner, allowing us to visualize the data (donut structure).

However, when we increase the depth to 2, the HVT function performs a more detailed variance decomposition by breaking down the overall variance into subcomponents at two levels. This increased level of analysis introduces additional dimensions or factors that are not easily represented in a 2D visualization.

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data at level 2 for better visualization.

metric_list <- colnames(hvt.torus3[[3]]$summary)
metric_list <- metric_list[7:9]
hmap <- list()
hmap <- lapply(1:length(metric_list), function(x){
muHVT::hvtHmap(
  hvt.torus5,
  torus_df,
  child.level = 2,
  hmap.cols = metric_list[[x]],
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 1,
  show.points = T,
  quant.error.hmap = 0.1,
  n_cells.hmap = 26
)
})

The heatmaps displayed below provides a visual representation of the spatial characteristics of the torus, allowing us to observe patterns and trends in the distribution of each of the features (X,Y and Z). The sheer green shades highlight regions with higher coordinate values in each of the heatmaps, while the indigo shades indicate areas with the lowest coordinate values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the torus structure

grid.arrange(hmap[[1]], nrow = 1, ncol=1)

grid.arrange(hmap[[2]] ,nrow = 1, ncol=1)

grid.arrange(hmap[[3]], nrow = 1, ncol=1)

7.1 Map A : Base Compressed Map

As we are familiar with the structure of the computers data, we will now follow the following steps to get the predictions using the Computers dataset.

Step 1: Data Compression

For more detailed information on Data Compression please refer to section 2 of this vignette.

Let us try to understand the HVT function first.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  projection.scale,
  normalize = T,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  diagnose = TRUE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of HVT function have been explained below :

• dataset - A dataframe with numeric columns

• min_compression_perc - An integer indicating the minimum percent compression rate to be achieved for the dataset

• n_cells - An integer indicating the number of cells per hierarchy (layers)

• depth - An integer indicating the number of layers. (1 = No hierarchy, 2 = 2 layers, etc …)

• quant.error - A number indicating the quantization error threshold. A cell will only breakdown into further cells if the quantization error of the cell is above the defined quantization error threshold

• distance_metric - The distance metric can be L1_Norm or L2_Norm. L1_Norm is selected by default. The distance metric is used to calculate the distance between an n dimensional point and centroid. The user can also pass a custom function to calculate this distance

• error_metric - The error metric can be mean or max. max is selected by default. max will return the max of m values and mean will take mean of m values where each value is a distance between a point and centroid of the cell. Moreover, the user can also pass a custom function to calculate the error metric

• quant_method - The quantization method can be kmeans or kmedoids. kmeans is selected by default

• normalize - A logical value indicating whether the columns in your dataset need to be normalized. Default value is TRUE. The algorithm supports Z-score normalization

• diagnose - A logical value indicating whether user wants to perform diagnostics on the model. Default value is TRUE.

• hvt_validation - A logical value indicating whether user wants to holdout a validation set and find mean absolute deviation of the validation points from the centroid. Default value is FALSE.

• train_validation_split_ratio - A numeric value indicating train validation split ratio. This argument is only used when hvt_validation has been set to TRUE. Default value for the argument is 0.8

First we will perform HVT using the computers dataset along with below mentioned model parameters to generate map A and try to achieve data compression of atleast 80%. In situations where the compression ratio does not meet the desired target we will iteratively increase the number of cells until the desired compression percentage is reached.

In this example usage we will compress the data at depth 1 as it provides a simplified overview of the data by calculating the overall variance and potentially reducing the dimensionality of the data

Model Parameters

• Number of Cells at each Level = 1001
• Maximum Depth = 1
• Quantization Error Threshold = 0.1
• Error Metric = Max
• Distance Metric = Manhattan

set.seed(240)
hvt.results <- list()
map_A <- muHVT::HVT(trainComputers,   
                          n_cells = 1001,
                          depth = 1,
                          quant.err = 0.1,
                          projection.scale = 10,
                          normalize = F,
                          distance_metric = "L1_Norm",
                          error_metric = "max",
                          quant_method = "kmeans",
                          diagnose = F)

Now let’s check the compression summary. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

compressionSummaryTable(map_A[[3]]$compression_summary)

As it can be seen from the table above, 83% of the cells have hit the quantization threshold error. Since we are successfully able to compress 83% of the data, so we will not further subdivide the cells

map_A[[3]] gives us detailed information about the hierarchical vector quantized data.

map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A

summaryTable(map_A[[3]]$summary)